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Second-Order Finite Difference Schemes

The simplest (and traditional) way of discretizing the 1D wave equation is by replacing first derivatives by first-order differences

\begin{eqnarray*}
\left.\frac{\partial y}{\partial t}\right\vert _{x=mX,t=nT} &\simeq& \frac{y_{m}^{n}-y_{m}^{n-1}}{T}\\
\left.\frac{\partial y}{\partial x}\right\vert _{x=mX,t=nT} &\simeq& \frac{y_{m}^{n}-y_{m}^{n-1}}{X}
\end{eqnarray*}

and second derivatives by second-order differences

\begin{eqnarray*}
\left.\frac{\partial^{2} y}{\partial t^{2}}\right\vert _{x=mX,t=nT} &\simeq& \frac{y_{m}^{n-1}-2y_{m}^{n}+y_{m}^{n+1}}{T^2} \\
\left.\frac{\partial^{2} y}{\partial x^{2}}\right\vert _{x=mX,t=nT} &\simeq& \frac{y_{m-1}^{n}-2y_{m}^{n}+y_{m+1}^{n})}{X^2}
\end{eqnarray*}

and so on, where $ y_{m}^{n} \mathrel{\stackrel{\mathrm{\Delta}}{=}}y(nT,mX)$ are the grid variables. Note that we have uniformly sampled the time-space plane, with timestep $ T$ (the temporal sampling interval) and space step $ X$ (the spatial sampling interval).



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``Modal Synthesis of a Piano String'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2019-02-05 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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